Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines

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1 I.J. Intelligent Systems and Alications, 2016, 4, 1-17 Published Online Aril 2016 in MECS (htt:// DOI: /ijisa Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines Kawal Jeet D. A. V. College, Jalandhar, , India Renu Dhir and Paramvir Singh Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India {dhirr, Abstract Nature-insired algorithms are recently being areciated for solving comlex otimization and engineering roblems. Black hole algorithm is one of the recent nature-insired algorithms that have obtained insiration from black hole theory of universe. In this aer, four formulations of multi-objective black hole algorithm have been develoed by using combination of weighted objectives, use of secondary storage for managing ossible solutions and use of Genetic Algorithm (GA). These formulations are further alied for scheduling jobs on arallel machines while otimizing bi-criteria namely maximum tardiness and weighted flow time. It has been emirically verified that GA based multi-objective Black Hole algorithms leads to better results as comared to their counterarts. Also the use of combination of secondary storage and GA further imroves the resulting job sequence. The roosed algorithms are further comared to some of the existing algorithms, and emirically found to be better. The results have been validated by numerical illustrations and statistical tests. Index Terms Auxiliary archive, Black Hole algorithm, Genetic algorithm, Job scheduling, Nature-insired algorithm, Tardiness, Weighted flow time. I. INTRODUCTION Nature has always been a source of insiration for human beings. This is reflected by recent technological advances in various fields of science and engineering [1]. Bat, Firefly, Artificial Bee Colony, Cuckoo Search, Flower Pollination, Black Hole and Grey Wolf algorithms are few algorithms that have been develoed by taking insiration from nature. These algorithms are widely used for solving various engineering and manufacturing otimization roblems in various fields ranging from routing in wireless sensor networks to scheduling of jobs in manufacturing unit [2-11]. Such engineering otimization roblems are generally accomanied by multile and often conflicting objectives or criteria, and are quite challenging to be solved. Insired from various alications of nature-insired algorithms, we investigated the usage of nature-insired Black Hole algorithm for otimizing scheduling of jobs on arallel machine on the basis of criteria namely Maximum Tardiness (T max) and Weighted Flow Time (WFT). In engineering and manufacturing roblems, there are often job instances with different rocessing times, due dates and riorities. Parallel machine scheduling involves a scheduling environment with m identical, arallel machines. Each of the n given jobs must be rocessed non-reemtively on any one of the available machines. However, machines can rocess jobs unattended. The roblem of scheduling multile jobs on arallel otential machines by otimizing T max and WFT is one of the imortant roblems that have been reeatedly encountered by engineers. A. Migration from Single Objective to Multi-Objective Black Hole Algorithm When comared with single objective otimization, multi-objective otimization is quite cumbersome [12]. This is quite evident from their theory of aroach. In case of single objective otimization techniques, the solution is a single oint in the solution sace. On the other hand, in case of multi-objective otimization technique numbers of equally otimal solutions in the solution sace are identified. This set of multile otimal solutions is called Pareto front. This Pareto front is comosed of solutions with relationshis that define the sueriority of one solution over another. More formally, for cost minimization roblems, the relation between solutions is defined as follows: F( x1) F( x2), if g, fg( x1) fg( x2) where f g is the value of g th fitness function and F(x) is the overall fitness of the solution x; g varies from 1 to h where h is the number of objectives to be otimized This means solution x2 is better than x1 if it is better according to at least one of the individual fitness (objective) functions and no worse according to all of the rest. In other words, x2 dominates x1. The Pareto front is the set of elements that are not dominated by any ossible element of the solution sace. The Pareto front thus denotes the best results achievable.

2 2 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines Generally, a single objective otimization algorithm cannot be used directly as multi-objective otimization technique [13]. Various techniques have been devised to adat single objective aroaches to multile objective aroaches. One such technique is to reformulate the roblem under consideration as a single objective roblem. This is ossible by adding relative weights to the objectives and erforming weighted summation of multile objectives to make a single objective [14]. The aroach now follows single objective method of otimization to identify best solution with otimum weighted sum of objectives. Another way is to find set of equally otimal non-dominated Pareto front [15]. So, it has been observed that significant changes are required to modify single objective to multi-objective otimization technique. In this aer, multi-objective Black Hole algorithms have been develoed from single-objective aroach for otimizing bi-criteria namely maximum tardiness (T max) and weighted flow time (WFT) for the jobs to be scheduled on arallel machines. Black Hole [16] has been insired by black hole theory of the universe. This theory describes a black hole as a lace in sace where gravitational force is so strong that even light can t escae. Efficiency and erformance of roosed algorithms have been comared to existing state of art otimization algorithms. The results thus obtained are statistically tested and indicate that roosed algorithms significantly outerforms the existing algorithms for the roblem of scheduling jobs on arallel machines. The rime contributions of this research work are listed below. Formulation and investigation of the use of Black Hole algorithm as multi-objective otimization technique using weighted sum of objectives for the rocess of job scheduling on arallel machines on the basis of bi-criteria namely T max and WFT. Investigation of the imact of hybridizing multiobjective Black Hole algorithm with Genetic Algorithm (GA). Formulation and investigation of the alication of Black Hole algorithm as multi-objective otimization technique (using auxiliary archive for external storage of Pareto front solution) for the rocess of job scheduling on arallel machines on the basis of bi-criteria namely T max and WFT. Investigation of the imact of develoing multiobjective Black Hole algorithm by using auxiliary archive as well as GA. Comarison of roosed algorithms to that of existing Multi-Objective Genetic Algorithm (MOGA), Non-dominated Sorting Genetic Algorithm-II (NSGA-II) and Multi-objective Particle Swarm Otimization (MOPSO) algorithms with resect to job scheduling on arallel machines. The rest of aer is organized as follows. In Section II, literature related to the field of job scheduling on arallel machines using nature-insired algorithms has been summarized. Section III addresses the formulation of the roblem along with various assumtions, notations and fitness functions. Section IV describes the roosed multi-objective Black Hole algorithms. Section V describes the exerimental setu and assessment criteria. In Section VI, scheduling results of randomly generated samles are emirically and statistically investigated. In Section VII, threats to the validity of the roosed work have been discussed. The aer is concluded in Section VIII mentioning some future recommendations. II. LITERATURE REVIEW Scheduling of jobs on arallel machines has been an ucoming research toic in the ast coule of decades. A large number of engineers and researchers have been working to erform otimization of multile criteria for scheduling number of jobs on arallel machines. Various meta-heuristic aroaches have been develoed to solve this roblem. In this section some of these aroaches have been discussed along with the criteria and heuristic followed. In one of the works in the field of job sho scheduling, Ruiz-Torres et al. [17] minimized the number of late jobs and their average flow-time. They used simulated annealing and neighborhood search for this task. In another work [18], Rahimi-Vahedet et al. minimized the weighted mean comletion time and weighted mean tardiness using Multi-Objective Scatter Search (MOSS). Chang et al. [19] otimized roduction and marketing using Ant Colony Otimization (ACO) technique for solving the roblem of multi-stage job sho arallel machine scheduling. They also incororated the concet of order slitting for multi-stage scheduling of jobs. In another work [20], total comletion time and makesan are minimized while scheduling jobs on m machines taking into consideration the sequence deendent set u times of jobs and by using heuristics such as Shortest Sum of Setu, Processing and Removal Times (SSPRT), Shortest sum of Setu and Removal Times (SSRT) and Earliest Due Date (EDD) sequence. Berrichi et al. [21] used evolutionary genetic algorithms to minimize makesan and system unavailability while scheduling n jobs on m machines. The Pareto front thus obtained is aroximated to obtain the desired otimized schedules. Berrichi et al. [22] roosed multi-objective ACO based aroach to otimize roduction and maintenance scheduling. The goal of their aroach is to identify the best assignment of roduction tasks to machines along with the reduction in reventive maintenance (PM) eriods of the roduction system. Mazdeh et al. [23] used a meta-heuristic algorithm based on Tabu search to minimize the total tardiness and machine deteriorating cost. Hybrid genetic algorithm based on the concet of multile suboulations has been used by Rashidi et al. [24] to minimize the makesan and maximum tardiness

3 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines 3 while scheduling jobs on unrelated arallel machines, taking into consideration the setu times of jobs. Artificial immune systems have been investigated [25] for solving the roblem of hybrid flow sho based on otimizing makesan of the jobs to be scheduled. In order to schedule jobs on arallel identical machines, the use of Time Petri Net based aroach has also been investigated [26] while minimizing the makesan and balancing the load among the machines Li et al. [27] used discrete multi-objective Artificial Bee Colony (ABC) to otimize makesan, the total workload of machines and the workload of the critical machine. They investigated the imact of schedules on maintenance activities. They used efficient initialization scheme and otimal Pareto front for obtaining efficient schedules for the jobs under consideration. Motivated by the existing works on the use of natureinsired algorithms for job scheduling, we investigated the alication of nature-insired Black Hole algorithm for the cause of job scheduling on arallel machines while otimizing T max and WFT. III. DESCRIPTION OF JOB SCHEDULING PROBLEM The roblem of scheduling multile jobs on arallel otential machines for otimizing maximum tardiness and weighted flow time is one of the imortant roblems that have been encountered by engineers time and again. In order to describe the roblem, let there be n jobs that can work indeendent of each other on m otential arallel machines. Each job has a due date, rocessing time and weight (or riority). All jobs are available at time zero. Further it is also resumed that a job cannot re-emt other job at any oint of time. The roblem is to schedule given n jobs on m machines in such a way that T max and WFT are minimized. The number of ossible combinations of jobs (to formulate a schedule) that could be investigated is exonentially large and solution of such a roblem is NP-hard [23]. Sharma et al. [28] roosed a heuristic that is able to solve this NP-hard roblem. But as the size of roblem increases, the efficiency and effectiveness of this aroach reduces. As mentioned above, nature insired algorithms have the caability to solve such large sized roblems. The following assumtions, notations and decision variables have been drawn for the formulation of the roblem. Assumtions: The roblem draws following assumtion. A job could be rocessed on any machine. All jobs are available at time zero. Jobs can execute indeendent of each other. A job could not re-emt any other job. All the machines are always available. Notations:Following notations are used for the formulation of roblem of job scheduling on arallel machines. n: Number of jobs to be scheduled m: Number of otential arallel machines i: Job under consideration d i: Due date of the i th job where i =1,2,...,n j: Order of rocessing of i th job on a machine, where j =1, 2,...,n k: Machine to which a job has been allocated, where k = 1,2,...,m i: Processing time of i th job w i: Weightage of i th job. It may indicate riority or imortance of the job. a ik: Time at which machine k is allocated to i th job c i: Time of comletion of i th job Ti: Tardiness of i th job A. Objective/Fitness Function Maximum Tardiness (T max) is the tolerance time u to which tardy job is ermitted by the system. T max max T i, where i=1 to n WFT= Sum of weighted comletion times of the jobs Decision Variables, n wc i i i1 1, if job i is allocated to machine k on ostion j xijk 0, otherewise 1, if job i is assigned to machine k yik 0, otherewise 1,if job r immediately follows job i on machine k z irk = 0,otherwise It could be deduced that zirk =1 if xijk 1 and xr( j1) k 1 B. Mathematical Formulation The mathematical formulation for the given roblem is as follows: Minimise: T max=max T i m n n Minimise WFT=Min( x * w * c ) ijk k1 j1 i1 The constraints followed while scheduling jobs are described below. T max is greater than equal to difference in comletion time and due date for all the jobs. i i

4 4 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines T c d i max i i, All i jobs have been allocated to a machine k at some osition j. n m n x ijk i1 k1 j1 n Allocation time of rocess r is equal to comletion time of rocess i rovided r follows i on machine k. a z c rk irk * i Variables are either 0 or 1 (binary). xijk, yik, zirk 0,1 Job i is assigned to only one machine and that too only once. C. Alication m yik 1 where i 1, 2,..., n k1 One of the ossible alications of the roblem under consideration is scheduling of large number of rocesses on multile rocessors by the oerating system. In such a system, numbers of rocessors (CPU) (as machines) are available all the time in arallel and number of rocesses (jobs) are to be scheduled on these rocessors. Each of the rocess (job) has its own rocessing time and riority. The rocesses are to be scheduled on available arallel rocessors such that there is minimum tardiness as well as minimum weighted flow time for the rocesses. This could lead to increase in the throughut rate. Further, use of arallel rocessors enhances roduction as the work does not sto when some rocessors fail or maintenance occurs. IV. JOB SCHEDULING ON PARALLEL MACHINES USING BLACK HOLE ALGORITHM Algorithms imitating rocesses found in nature are called nature-insired algorithms. It has been a field of inclination for eole in scientific and engineering studies in recent years. In actual engineering roblems, different objectives often conflict with each other, and hence obtaining an otimal result is very difficult. Natureinsired algorithms are found to be very efficient in solving such comlex roblems. We roose to aly multi-objective Black Hole algorithm to the roblem of job sho scheduling on arallel machines taking into consideration the objectives namely T max and WFT. This section describes our adatation of nature-insired algorithms to the roblem of job scheduling on arallel machines. This adatation includes: reresentation of the solutions and the generation of the initial oulation; evaluation of individuals using multile objectives namely T max and WFT; selection of the individuals from one generation to another; generation of new individuals; bringing algorithm out of local otima using genetic oerators (to exlore the search sace in a better way); augmenting auxiliary archive to maintain non-dominated Pareto front for future considerations. A. Solution Reresentation / Problem Encoding The meta-heuristic algorithms roosed in this aer need aroriate encoding of the roblem. It is observed that the encoding of the oulation has an imact on the quality of results generated by a meta-heuristic algorithm. In this roosed work, we encoded oulation as individuals made of floating oint numbers. Size of each individual is equal to the number of jobs to be scheduled. Each osition in an individual is reresented by a decimal number having an integer art to indicate the machine number to which job has been allocated and a decimal art to indicate the order in which this job is allocated to that machine. As an examle, let there be 5 jobs to be scheduled on two arallel machines. One of the individuals in the oulation is encoded as Job id Individual This encoded individual has five ositions. For job 1, the value of allele is It means job1 is allocated to Machine 2 at the front followed by job 3 with allele value B. Generation of Candidate Poulation To generate the oulation, following stes are followed. 1. Randomly allocate all n jobs to m machines using (1). X rand (1, m) (1) j Where =1,2,,Po (Number of candidate solutions to be initially generated); Po is the size of oulation to be used (as shown in Table 1); i=1,2,,n where n is number of decision variables to be otimised (the number of jobs to be scheduled); m is the number of machines available. The value acts as a characteristicof the floating tye decision variables. For each individual in the oulation, ReeatSte 2 2. For k=1 to m machines //Find all the jobs allocated to machine k Y Find ( X i k )

5 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines 5 //In order to find the order of allocation of jobs on machine k, create ermutation of jobs allocated to the machine k. Index=Permutation(Y ) //Jobs are allocated to machine k on the basis of their ermutation (order in index). This value is used as mantissa for the floating tye decision variables (as discussed above) and the resulting decision variable becomes X i. j and xijk 1 C. Proosed Job Scheduling Algorithms This section rooses four stewise algorithms that are based on the chosen nature-insired Black Hole algorithm, and its hybrids with GA and auxiliary archive. All these algorithms are used to solve the roblem of job scheduling on arallel machines. The first algorithm is based on multi-objective formulation of Black Hole algorithm based on weighted objectives. T max and WFT have been used as objective to be otimized. The second algorithm is develoed by hybridising the first algorithm with GA. The third algorithm uses auxiliary archive to store non-dominated Pareto front. The fourth algorithm has been develoed by adding Genetic algorithm to the third algorithm. Following section describes the roosed algorithms in detail. Multi-Objective Weighted Black Hole Algorithm (MOWBH): In this algorithm, Black Hole algorithm is imlemented as multi-objective aroach using weighted objective sum technique. In this technique, all h objectives under consideration are combined to make a single objective (F) using (2) [29]. Each of the objectives is assigned random weights such that h h g 0, g 1 and F= g * g (2) g1 g1 w w w f g indicates the objective under consideration, g varies from 1 to h (number of ojectives to be otmized), w g is the random weight allocated to g th objective, f g is the value of g th objective and F is the combined single objective calculated from all the objectives in the roblem under consideration. The basic structure of this algorithm is as discussed below. Ste 1 [Initialize oulation]: Encode and initialise the oulation of candidate solutions as discussed above. The detail of common arameters to be used in the imlementation of these algorithms is given in Table 1. These values are obtained by manual tuning after reeated executions of the algorithms. Evaluate fitness of each candidate in the oulation on the basis of combined weighted objective F (2). Designate the solution with least value of F as Black Hole (X BH). Table 1. Common control arameters for Black Hole algorithm for job scheduling on Parallel machines Parameter Value Descrition Number of variables to be otimised (n) Poulation size Number of jobs to be sequenced 5 times the number of jobs to be scheduled Poulation Candidate schedules for the jobs Generations 15 times the number of jobs to be scheduled (n) and number of machines to be used in arallel(m)= 15*m*n Or When solution does not change for 400 consecutive generations Objective T max and WFT functions Lower bound for decision variables i ( X min ) Uer bound for decision variables i ( X max ) Size of REP Number of decision variables in the individual is equal to the number of jobs to be sequenced. With increase in number of jobs to be scheduled, oulation size increases. Randomly generated. Stoing criteria Weighted sum of objectives (in MOWBH and MOWBHGA) or Pareto front of ossible nondominated individuals (in MOBH and MOBHGA) is created using these objectives deending on the algorithm used. 1.1 Job is allocated to machine number 1 before any other job. m.n 1% of size of oulation Job is allocated to last machine m and in worst case all n jobs are allocated to this last machine. In this way, order of job will be n. To kee track of best (nondominating) solutions Reeat Stes 2-4 until stoing criteria is met (as indicated in Table 1). Ste 2 [Identify new ossible solutions]: For each iteration t, identify new location (X (t+1)) for each job sequence (X (t)) by using (3) X ( t 1) X ( t ) rand *( X BH X ( t )) (3) Ste 3 [Search for a better solution]: Calculate the fitness of this new solution X (t+1) by using (2). If new candidate solution is better than the current candidate solution, then relace the current solution with this new solution else ignores it. This ste is required to locally

6 6 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines search for better sequence. It moves the current candidate randomly in search for a better solution. If the new solution is better the current Black Hole (X BH), then designate this new solution as new Black Hole. Ste 4 [Discard vanished solution]: Calculate the radius of event of horizon of the Black Hole job sequence by using (4). R F BH o 1 wheref BH is the fitness for Black Hole job sequence and F is the fitness of th job sequences calculated using weighted fitness function calculated using (2). Po is the size of oulation under consideration (as mentioned in Table 1). If a star enters this event horizon, it is absorbed by the Black Hole. If (F - F BH < R), the job sequence is discarded as it is regarded to be entered in event horizon of Black Hole and is thus vanished. Generate new job sequence to balance the size of oulation. Ste 5 [Outut]: Outut the best job sequence. Black Hole algorithm works to find otimized results but might get stuck at local otima. GA has an inherent caability of erforming a wider search ability due to the use of crossover and mutation oerators. The crossover oerator hels in swaing of information between two candidate solutions which increases the ability to investigate new solution areas, whereas the mutation oerator increases the diversity of the oulation and hels to avoid the local otima. Multi-Objective Weighted Black Hole Genetic Algorithm (MOWBHGA): In this algorithm, genetic hase has been aended to MOWBH after Ste 4 to develo MOWBHGA. Combined fitness function (2) used in MOWBH is taken as the objective to be otimized. New Ste 5 (Genetic Phase) is described as below: Ste 5.1[Inut]: Take oulation of candidate job sequences from Ste 4 of MOWBH as inut oulation of chromosomes. Calculate fitness of each solution using combined fitness function F (using (2)). Ste 5.2 [Selection]: Select two arent chromosomes (ossible job sequence) from the oulation using tournament selection on the basis of their fitness calculated in Ste 5.1. Ste 5.3 [Crossover]: Cross the arents selected in Ste 5.2 to create new children with crossover robability of 0.6. Arithmetic crossover oerator has been used for this urose. The crossover robability can be obtained by reeated executions of the algorithm and has been tested manually. F (4) Ste 5.4 [Mutation]: Mutate new child at random ositions (job) in the chromosome with mutation robability of 0.2. Ste 5.5 [Relace]: If this new offsring is better than the arents (lesser fitness value) then accet it. The oulation thus obtained is used by next iteration of Black Hole algorithm. The arameters to be used to imlement GA are shown in Table 2. These arameters are obtained by manual tuning and reeated executions of the algorithms under consideration. Table 2. Control arameters for GA Parameter Value Descrition Selection algorithm Tournament (Size 2) Candidate sequences Parent1 and Parent2 are selected for crossover using this method. It could be efficiently coded, works on arallel architectures and allows the selection ressure to be easily adjusted Crossover function Mutation function Objective functions Arithmetic Uniform T max and WFT Child=R1 * Parent1+ R2 * Parent2 Where R1,R2 are indeendent random numbers between 0 and 1 It always roduces feasible offsring for a convex solution sace. Set crossover fraction to 0.6. Set mutation rate to Manually tested to give best result. Weighted sum of objectives (in Proosed Algorithm 2) or Pareto front of ossible non-dominated individuals (in Proosed Algorithm 4) is created using these objectives deending on the algorithm used. Multi-Objective Black Hole Algorithm (MOBH): The algorithm discussed above have a serious shortcoming. They return the single objective and are highly deendent on the weights. At the same time allocation of weights to the individual objective is difficult. In case of random selection of weights, the outut is highly unredictable. So, we investigated the use of Pareto otimization [30] (resulting in solutions in the form of Pareto front) for otimization of job schedule using Black Hole algorithms discussed above. In order to store this Pareto front, an auxiliary archive and hyercubes are used to select best solutions (MOBH) for each iteration of the algorithm. Ste wise imlementation of this algorithm is discussed below. Ste 1 [Initialize oulation]: Encode and initialize the oulations of individuals as discussed above. The detail of common arameters to be used in the imlementation of these algorithms is given in Table 1. Ste 1.1: Evaluate fitness of each candidate in the oulation using objective functions namely T max and WFT. Ste 1.2: Store the job sequence that reresent nondominated vectors in the temorary reository also called auxiliary archive and is named REP. Ste 1.3: Generate hyer-cubes of the search sace exlored so far, and locate the sequence using these

7 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines 7 hyer-cubes as a coordinate system where coordinates of each job sequence are defined according to the values of its objective functions [31]. Ste 1.4: Select the best solution achieved so far and designate it as Black Hole (X BH). Reeat Stes 2 to 4 for a redefined number of iterations (as shown in Table 1). Ste 2 [Identify new ossible solution]: For each iteration t, identify new location (X (t+1)) for each job sequence (X (t)) by using (3). Each of the i th decision variables (X i ) of the current candidate solution () is ket within ermitted uer and lower bound (as shown in Table 1) for the job sequence under consideration. Ste 3 [Search for a better solution]: Calculate the fitness of this new solution taking into consideration the non-dominance of the job sequences. If this new solution dominates the current candidate solution, then relace candidate solution with this better new solution. This ste is required to search locally for a better job sequence. It moves the current candidate randomly in search for a better solution. If the new solution dominates the current Black Hole (X BH) then designate this new solution as new Black Hole. Ste 4 [Discard vanished solution]: For calculating radius of event of horizon (of Black Hole) in nondominated Pareto front, calculate comonents of radius on the basis of objectives calculated using (5). For all h objectives Rh ( ) f BH o 1 f ( h) ( h) where f BH is the fitness value of the Black Hole job sequence and f (h) is the fitness value of the h th objective of th job sequence. For the roblem of job sequence under consideration, the number of objectives (h) is equal to 2. For each individual in the oulation, if difference in fitness value of every objective function and Black Hole dominates corresonding comonent of R (f (h)-f BH dominates R(h)) calculated in (5), the job sequence is discarded as it is regarded to be entered in event horizon of Black Hole and is thus vanished. Generate new job sequence to balance the size of oulation. Udate REP as well as the geograhical reresentation of job sequences within hyer-cubes [32] by inserting all the currently non-dominated locations into the auxiliary archive (REP) and eliminating dominated individuals. Ste 5 [Outut]: REP contains the ossible Pareto front i.e. the ossible non-dominated solutions as outut. (5) Multi-Objective Black Hole Genetic Algorithm (MOBHGA): Similar to shortcoming of MOWBH algorithm, MOBH algorithm might get stuck at local otima. Due to the reasons mentioned for MOWBHGA, GA is aended as Ste 5 in MOBH to develoed MOBHGA algorithm. New Ste 5 (Genetic Phase) is described as below: Ste 5.1 [Inut]: Take the oulation of candidate job sequence from the nature-insired algorithm under consideration as inut oulation of chromosomes. Calculate the fitness of each solution using the nondominance aroach on the basis of objectives namely T max and WFT. Ste 5.2 [Selection]: Select two arent chromosomes (ossible job sequences) from the oulation using tournament selection on the basis of their fitness calculated in Ste 5.1. Ste 5.3 [Crossover]: Cross the arents selected in Ste 5.2 to create new children with crossover robability of 0.6. Arithmetic crossover oerator has been used for this urose. The crossover robability is obtained by reeated executions of the algorithm, and has been tested manually. Ste 5.4 [Mutation]: Mutate new child at random ositions (assignment of a job to a machine is changed) in the chromosome with mutation robability of 0.2. Ste 5.5 [Relace]: If this new offsring is better than the arents (non-dominating), then accet it. The arameters to be used in this hase are shown in Table 2. V. EXPERIMENTAL SETUP This section describes the exerimental set u for imlementation of roosed algorithm and their analysis. The comarison of the roosed nature-insired multiobjective Black Hole algorithms for job sho scheduling on arallel machines has been done with some of the already known search based algorithms: MOGA [33], NSGA-II [34] and MOPSO [31] algorithm. Table 3. Detail of arameters required to generate random samles. Parameter Value Number of Jobs (n) 40,60 Number of Machines 2,3,6 (m) Uer bound for 80 rocessing i time( X max ) Lower bound for i rocessing time( X min ) Weight/Priority of the job (w i) Due date for the job (d i) 20 Randomly generated numbers between lower bound and uer bound Randomly generated and greater than rocessing time for the job

8 8 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines A. Samles Under Study All algorithms considered in this aer are imlemented in MATLAB 8 [35]. The multi-objective algorithms discussed in this aer are alied on samles generated randomly in MATLAB. The number of arallel machines to be used for scheduling is taken as 2, 3 or 6. The uer and lower bound along with other details required for generating random samles are given in Table 3. The algorithms under consideration are search based and are highly randomized, so these are reeated 30 times and best job sequence is taken as the outut. B. Assessment Criteria In order to assess the quality of job sequence obtained by the alication of roosed algorithms, Total Cost (sum of T max and WFT), Pareto front (T max vs WFT) and evolution of Total Cost with number of generations have been used as assessment criteria. C. Total Cost of Job Sequence as Assessment Criteria In order to evaluate the quality of resulting job sequences, objectives T max and WFT are used. All algorithms considered in this aer are multi-objective in nature, so they lead to Pareto-front as an outut. Since Pareto front consists of non-dominated ossible solutions, where none of the solutions in Pareto front dominates others, we used Total Cost (sum of T max and WFT) of the solution as criteria to assess the quality of job sequence. Lower the value of this cost better is the job sequence solution. D. Pareto Front as Assessment Criteria In order to comare the kind of non-dominated solution thus obtained, we used Pareto front obtained by the alication of all algorithms under consideration. The Pareto front is obtained by lotting T max on X-axis and WFT on Y-axis. E. Evolution of Objectives with Generations as Assessment Criteria The change in the values of objectives with generations/iterations are used to comare the stability (number of generation it needs to give a constant value for the objective) of the algorithms being used. F. Statistical Validation Since meta-heuristic algorithms considered in this aer are search based and are highly randomized otimizers, they may rovide different results for the same roblem instance from one execution to another. Hence, the algorithms are run 30 times. In order to statistically validate the significantly better results obtained by the use of GA (as one of the hases of the algorithms) as well as the use of auxiliary archive, the Wilcoxon rank sum test [36] with a 95% confidence level (α = 5%) has been conducted. The tests have been conducted between MOWBH and MOWBHGA; MOBH and MOBHGA; MOBHGA and MOGA; MOBHGA and NSGA-II; MOBHGA and MOPSO. The null and alternative hyothesis for the same is as discussed below. Null hyothesis H0: The obtained results by the alication of algorithms under consideration are samles from continuous distributions with equal medians. Alternative Hyothesis H1: The obtained results by the alication of algorithms under consideration are not samles from continuous distributions with equal medians. The test returns h-value and -value. The h-value indicates whether the null hyothesis H0 is acceted or rejected. The -value indicates the robability of rejecting the null hyothesis H0 while it is true (tye I error). A - value that is less than or equal to α ( 0.05) means H1 is acceted. The corresonding h-value is 1. If the -value is strictly greater than α (> 0.05), Alternative Hyothesis H1 is rejected and Null Hyothesis H0 is acceted. The corresonding h-value is 0. VI. EXPERIMENTS RESULTS The multi-objective algorithms discussed in this aer are alied on samles of varied sizes generated randomly on the basis of arameters shown in Table 3. The quality of resulting job sequences are assessed on the basis of criteria as discussed below. A. Total Cost of Job Sequence as Assessment Criteria Table 4 shows the Total Cost of non-dominated job sequence with lowest total of T max and WFT criteria. The job sequences are obtained by the alication of MOWBH, MOWBHGA, MOBH and MOBHGA for all the samles under consideration. In order to further comare our findings, we lot the Total Cost (T max and WFT) obtained by the alication of all the algorithms and for all the samles. Fig. 1 lots Total Cost obtained by alication of MOWBH and MOWBHGA algorithms for all the samles under consideration. Fig. 2 lots Total Cost obtained by the alication of MOBH and MOBHGA algorithms for all the samles under consideration. By close evaluation of Tables 4 and Fig. 1, it is observed that the hybrid algorithm obtained by the alication of GA (MOWBHGA) outerforms MOWBH. Similar imrovements in quality of job schedules have been observed (in terms of Total Cost) by using GA with MOBH (to obtain MOBHGA) as observed by analysis of Table 4 and Fig. 2. This imrovement in efficiency of MOWBH and MOBH is due to the use of GA which tends to bring these algorithms out of local otima and hence directs the search towards global otimization. Fig. 3 dislays Total Cost obtained by the alication of MOWBHGA, MOBHGA and comares the results to that of existing MOGA, NSGA-II and MOPSO algorithms. Analysis of Table 4 and Fig. 3 reveals that MOBHGA that is based on the use of auxiliary archive and GA erform better than MOWBHGA.

9 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines 9 Further analysis shows that MOBHGA outerforms all other existing algorithms (MOGA, NSGA-II and MOPSO) in terms of Total Cost of the job sequence thus obtained. B. Pareto Front as Assessment Criteria Fig. 4-9 lots Objective 1 (T max) on X-axis and Objective 2 (WFT) on Y-axis (for each algorithm) for all the samles under consideration. It shows the kind of solution sace thus obtained. In a multi-objective environment, it is called Pareto front which shows the set of non-dominated solutions sace. On analysis of Fig. 4-9, it is deduced that MOGA results in dense Pareto front as comared to the roosed algorithms. But if we select the solution with least values of both T max and WFT, it can be easily seen that MOBHGA outerforms other algorithms including MOGA. C. Evolution of Objectives with Generation as Assessment Criteria Fig lot the fitness function (Total Cost) and generations (iterations) obtained by the alication of all the roosed algorithms for all the samles under consideration. If we analyse this evolution, it is observed that although MOGA gets stable in lesser iterations as comared to other algorithms, MOBHGA is found to result in better otimizations in most of the cases (excet for samle with 40 jobs to be scheduled on 2 machines in which MOGA results in Total Cost comarable to MOBHGA). For 4 out of 6 samles, MOWBHGA is found to otimize better than MOGA. MOWBHGA and MOBHGA are found to get stable in lesser iterations as comare to NSGA-II and MOPSO in all the cases. D. Statistical Test Wilcoxon rank sum testhas been conducted between MOWBH-MOWBHGA; MOBH-MOBHGA; MOBHGA-MOGA; MOBHGA-NSGA-II and MOBHGA-MOPSO (for all the samles under study). The results thus obtained are shown in Table 5. On analysing Tables 4, it is observed that adding GA to MOWBH and MOBH imroves the rocess of job scheduling. Values of Wilcoxon rank sum test further validates the findings and justify the observations that in most of the cases, h-values corresonding to Wilcoxon test between MOWBH-MOWBHGA and MOBH- MOBHGA are 1 which indicates that the samles are from distributions having different medians. It means the sequences obtained by the alication of GA based algorithms are significantly better. Analysis of Table 4 reveals that MOBHGA leads to best job sequences for the roblem under consideration. Table 5 reveals the difference to be significant as h-value obtained as a result of Wilcoxon rank sum test between MOWBHGA and MOBHGA is 1 in all the cases. So, further Wilcoxon rank sum tests have been erformed between MOBHGA and state of art otimization algorithms. If we analyse the results of test between MOBHGA- MOGA; MOBHGA-NSGA-II and MOBHGA-MOPSO, it is observed that in case of MOPSO, the h-value is 1 in all the cases, in case of NSGA-II, the h-value is 1 in all the cases excet for 40 jobs to be scheduled on 6 machines; whereas in case of MOGA, the values are 1 in most of the cases excet for 40 jobs to be scheduled on 3 machines and for 60 jobs to be scheduled on 2 machines. So, it is statistically validated that MOBHGA leads to otimum job sequences as comared to other algorithms in most of the cases. Number of Machines Algorithm Number of Jobs Table 4. Solution with least Total Cost for randomly generated samles Value of MOWBH MOWBHGA MOBH MOBHGA MOGA NAGA-II MOPSO Objective T max WFT Total Cost T max WFT Total Cost T max WFT Total Cost T max WFT Total Cost T max WFT Total Cost T max WFT Total Cost

10 10 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines Fig.1. Best values obtained with 40 and 60 jobs scheduled on 2, 3 and 6 machines by MOWBH and MOWBHGA algorithms Fig.2. Best values obtained with 40 and 60 jobs scheduled on 2, 3 and 6 machines by MOBH and MOBHGA algorithms Fig.3. Best values obtained with 40 and 60 jobs scheduled on 2, 3 and 6 machines by MOWBHGA, MOBHGA, MOGA, NAGA-II, MOPSO algorithms

11 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines 11 Fig.4. T maxvs WFT (Pareto Front) for all the algorithms for 40 jobs scheduled on 2 machines Fig.5.T maxvs WFT (Pareto Front) for all the algorithms for 40 jobs scheduled on 3 machines Fig.6. T maxvs WFT (Pareto Front) for all the algorithms for 40 jobs scheduled on 6 machines

12 12 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines Fig.7. T maxvs WFT (Pareto Front) for all the algorithms for 60 jobs scheduled on 2 machines Fig.8.T maxvs WFT (Pareto Front) for all the algorithms for 60 jobs scheduled on 3 machines Fig.9.T maxvs WFT (Pareto Front) for all the algorithms for 60 jobs scheduled on 6 machines

13 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines 13 Fig.10. Evolution of Total Cost based on number of generations for 40 jobs scheduled on 2 machines Fig.11. Evolution of Total Cost based on number of generations for 40 jobs scheduled on 3 machines Fig.12. Evolution of Total Cost based on number of generations for 40 jobs scheduled on 6 machines

14 14 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines Fig.13. Evolution of Total Cost based on number of generations for 60 jobs scheduled on 2 machines Fig.14. Evolution of Total Cost based on number of generations for 60 jobs scheduled on 3 machines Fig.15. Evolution of Total Cost based on number of generations for 60 jobs scheduled on 6 machines

15 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines 15 Table 5. Wilcoxon test between roosed and state of art algorithms Machines=2 Machines=3 Machines=6 Algorithm Test Value N=40 N=60 N=40 N=60 N=40 N=60 MOWBH Vs MOWBHGA MOBH Vs MOBHGA MOWBHGA Vs MOBHGA MOBHGA Vs MOGA MOBHGA Vs NSGA-II MOBHGA Vs MOPSO Tmax WFT Tmax WFT Tmax WFT Tmax WFT Tmax WFT Tmax WFT h e e e e e h e e e e e h e e e e e e e h e e e e e e e e-004 h e e e e e e e e e-004 h e e e e e e e e e e e e e e e-031 E. Summary of Results By analysis of all the above assessment criteria, it is emirically evaluated that MOWBHGA erforms better than MOWBH. Similar imrovement has been encountered for MOBHGA which is otimizing better than MOBH. It is emirically evident that MOBHGA erforms better than other state of art algorithms: MOWBHGA, MOGA, NSGA-II and MOPSO. The statistical test further validates the imrovement to be significant. VII. THREATS TO VALIDITY Threats to validity refer to the factors that can bias our emirical study. With resect to current research work, there are two kinds of threats; internal and external validity. Internal validity refers to the biases in the results related to roosed algorithms. We take into consideration the internal threats to validity while tuning of the arameters. In order to mitigate this threat, we reeated the executions of algorithms and manually tuned the arameters involved. External validity refers to the generalizations that are done beyond the samles used in emirical study. In this study, we mitigated external threat by erforming exeriments on randomly generated samles. For generalization of results, more exeriments need to be erformed. VIII. CONCLUSIONS Scheduling jobs on arallel machines, while taking care of maximum tardiness and weighted flow time, is one of the issues that has been encountered in various engineering and manufacturing roblems and has found a large number of alications. A large number of heuristic and meta-heuristic techniques have been develoed to solve this roblem. In this aer, weighted objective multi-objective Black Hole algorithm along with its hybrid with GA has been roosed for solving such scheduling roblems. Further, multi-objective Black Hole is develoed by using auxiliary archive (to kee track of non-dominated solutions) and GA. This algorithm yields areciable results with a significant imrovement in job scheduling rocess. The results are further verified emirically by erforming numerical illustrations and Wilcoxon test. The work can be further extended by exerimenting with other novel nature-insired algorithms such as Bat [14] and Firefly algorithms[11]. Other criteria such as number of tardy jobs, earliness, tardiness etc. could also

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17 Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines 17 otimization," Evolutionary Comutation, IEEE Transactions on, vol. 8, , [32] J. D. Knowles and D. W. Corne, "Aroximating the nondominated front using the Pareto archived evolution strategy," Evolutionary comutation, vol. 8, , [33] K. Deb, Multi-objective otimization using evolutionary algorithms, vol. 16. New York: John Wiley & Sons, [34] K. Deb, A. Prata, S. Agarwal, and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," Evolutionary Comutation, IEEE Transactions on, vol. 6, , [35] In.mathworks.com. (2014, 16 December 2013). MATLAB - The Language of Technical Comuting - B. [36] D. J. Sheskin, Handbook of arametric and nonarametric statistical rocedures, 4 th ed., London: Crc Press, 2003, How to cite this aer: KawalJeet, RenuDhir, Paramvir Singh,"Hybrid Black Hole Algorithm for Bi-Criteria Job Scheduling on Parallel Machines", International Journal of Intelligent Systems and Alications (IJISA), Vol.8, No.4,.1-17, DOI: /ijisa Authors Profiles KawalJeet is an Assistant Professor in Post-Graduate Deartment of Comuter Science, D.A.V. College, Jalandhar, India. She received her Master s of Technology in Comuter Science from Dr. B.R. Ambedkar National Institute of Technology, Jalandhar, India in Currently, she is ursuing Ph.D from this institute. Her current research interest focuses on nature-insired comutation, software modularization, Bayesian networks, and software quality. She has ublished her research work in more than 20 international journals and conference roceedings. She is a member of the IRED, UACEE and ACM India. Dr. RenuDhir is working as Associate rofessor and Head in the Deartment of Comuter Science and Engineering National Institute of Technology, Jalandhar (Punjab), India. She has done M. Tech. in Comuter Science and Engineering from TIET Patiala, India, in 1997 and Doctor of Philosohy (Ph.D.) in Comuter Science and Engineering, Punjabi University, Patiala, India, in March She has ublished her research work in more than 40 International / National Conferences and Journals in various fields like Information Security, Image Processing, OCR, NLP and Pattern recognition. Dr. Paramvir Singh is working as Assistant Professor in Deartment of Comuter Science and Engineering, National Institute of Technology, Jalandhar (Punjab), India. He received the Ph.D. degree in Comuter Science and Engineering from Guru Nanak Dev University, Amritsar (Punjab), India, in 2011 and the M.Tech. degree in Comuter Science and Engineering from Punjab University Chandigarh, India, in He has ublished his research work in refereed international journals and conference roceedings. He is a member of the IEEE and the IEEE Comuter Society, and a life member of ISTE. His research interests include software engineering, object-oriented rogramming and data structures.